Theory for Free Rolling of Ship

Abstract

This paper deals with the free rolling of ship theoretically and experimentally, and the following results are obtained.(1) The free rolling of ship is made up of coupled oscillations of a horizontal transverse motion of its centre of gravity G with a rotational motion of ship about a longitudinal axis through G. And the horizontal motion is made up of two motions, the one being a damped oscillation and the other a damped translation, but the rotation simply a single damped oscillation. That is to say that, G, about the longitudinal axis through which t1 ship is rotating, translates transversely in the direction against the initial heeling, oscillating horizontally, and at last after the shift of a certain distance the motion is at rest.(2) C0, the apparent centre of free rolling, which is the point having a minimum double amplitude of horizontal oscillation 2|ξ|m0 among the points in the middle line of the ship, is at a distance Y0 below G. For box-shaped model ship, a ratio Y0/d, where d is a draught of ship, has a value of about 0.08, and a ratio 2|ξ|m0/d increases as the initial angle of heel θ0increases and has a value of 0.03 at θ0=35°. And these calculated values of Y0and 2|ξ|m0 are in agreement with the experimental results.(3) The actual amplitudes of horizontal oscillations of any point in the middle line_of the ship are larger and more irregular than the calculated ones, but The direction and the amount of the horizontal translation calculated agree with the experimental results AndD, a calculated distance of a horizontal shift of G between the time elapsed from the initial and the final instant of the motion, coincides perfectly with the experimental results, andD/dincreases as θ0increases and has a value of 0.9 at θ0=35°.(4) A damping power oe and α circular frequency ω in the free rolling of the ship are expressed as follows.α=α0-Ca, ω=√ω02+Cω, where α0and ω0are a damping power and a circular frequency respectively in the free rolling of the ship, the horizontal motion of the centre of gravity of which are restrained. Therefore, Ca andCω are the effects of the horizontal motion on the free rolling, and the calculated Ca and Cω agree with the experimental results. For the box-shaped model ship, the calculated Ca/α, Cω/ω2 and (T0-T)/T (where T and T0 are π/ω and π/ω0 respectively) have the following values.Ca/α : 7%4% in the range of θ0=0°35°.Cω/υ2 : about 4% throughout the same range of θ0 as above.(T0-T)/T : about 2%, , , , .(5) If the effect of the horizontal motion on the free rolling should be neglected as small, the damping power α and the period T, which shall be greatly influenced by the rotational motion, are approximately expressed as follows.α_??_I10h1/2 (I+I1h1)T2_??_π2·(I+I1h1) /W·h·G3